goodman



(No Model.) 2 Sheets-Sheet 1. J. GOODMAN.

I PLANIMETER, No. 496,562. Patented May 2, 1893.

W Wmses:

FIG.

(No Model;) 2 Sheets-SheetZ.

J. GOODMAN.

PLANIMBTER No. 496,562 Patented May 2, 1893 WJZWESSE'S NITED STATESPATENT OFFICE.

JOHN GOODMAN, OF LEEDS, ENGLAND.

PLANIMETERQ SPECIFICATION forming part of Letters Patent No. 496,562,dated May 2, 1893.

Application filed September 9, 1891. Serial 110,405,201. (No model.)Patented in England January 9, 1891, No. 407; in Belgium To a. whom itmay concern.-

Be it known that 1, JOHN GOODMAN, a subject of the Queen of GreatBritain and Ireland, residing at Leeds, in the county of York, England,have invented new and useful Improvements in Planimeters, (for which Ihave applied for Letters Patent in Great Britain, No. 407, bearing dateJanuary 9, 1891; in France, No. 202,216, dated August 10, 1891, and inBelgium, No. 72,304, dated August 6, 1891,) of which the following is aspecification.

My invention refers to improvements in that description of instrumentknown as the Stang planimeter (Cornelius Knudsen, of Copenhagen, maker)which consists essentially of a light rod or beam, hereinafter referredto as the beam, carrying at one end a tracing point, and at the oppositeend a marker or hatchet, which are hereinafter referred to as the tracerand hatchet respectively. This hatchet has a blade or edge whichdisposes it to run in straight or slightly curved lines; lateral motionbeing resisted by the sides of the slight groove which it forms in thesurface of the paper or other material over which it runs. This edge ishowever, so rounded from end to end as to allow a certain amount ofpivotal motion.

The mode of using the Stang planimeter is as follows: The tracer isplaced as near as can be judged at the center of the figure whose areais to be measured, and a dent or impression is made upon the surface ofthe paper with the hatchet. The tracer is then moved away from theinstrument side of the figure in a straight line toa point on theboundaryline of the figure, and carried completely round and along theboundary line until reaching the starting point on the boundary line,and therefrom back to the center starting point along the previouslymentioned straight line. A second dent is then made with the hatchet.The area of the figure is a function of the distance between the twodents and the length of the beam; or more strictly of the angle includedbetween the first and final positions of the beam, and the length of thebeam. Now if the figure whose area is to be measured is inclosed in arectangle, and the length of the beam be made equal to the rance August10, 1891. No. 202,216.

length of the rectangle, then the distance between the hatchetdents'gives the mean height of the figure. In some cases, as for examplethe indicator diagrams from steam and gas engines, it is rather theaverage height than the area of the diagram which it is desired toobtain.

Now my principal improvement consistsin making the instrument of thesame length as the diagram, and as the lengths of the diagrams usuallyvary'l make either the hatchet or tracer adjustable on the beam. Or forthe same reason, the beam may be made in two parts the one capable ofsliding telescopically within or over the other.

To facilitate accurately adjusting the distance between the tracer andthe hatchet, the vertical edges (preferably the inner ones) of thetracer and hatchet legs may be formed so as to give exactly the distancebetween the .central longitudinal lines of the said tracer and hatchet.Such edges are formed at right angles to the beam.

I have further found that when the instrument is used as hereunder setforth, there is an almost constant error of two and a half per cent. inthe result, so that I make the edges of the hatchet and tracer, so thatthere is a difference between the actual length of the diagram and thelength of the beam to counteract the above error. Owing to thecomparatively small variation in the lengths of the diagrams, thisalteration of two and a half per cent. on the mean or usual length ofthe diagrams, is a sufficient correction for all' diagrams taken withthe same instrument (indicator).

The tracer or hatchet (when their distance apart has been adjusted) maybe fixed in position by means of ordinary clamping screws, butpreferably they are provided with a spring friction lever arrangement,so that while insuring sufficientfrictional resistance to prevent anyaccidental or unintentional alteration of the position of the tracer orhatchet, the pressure of the lever can be at once readily released forthe purpose of permitting adjustment by hand. a

Figure 1 is a longitudinal elevation of my improved planimeter,in whichthe hatchet and tracer sockets are shown in section. Figs.

2 and 3 are cross sections on lines A B and C D respectively of Fi 1.Fig. 4 is a diagram of an irregular figure with lines which illustratethe method of measuring the same. Figs. 5 and 6 are additional diagramsillustrating the mathematical computation embodied in the specification.Fig. 7 shows the planimeter applied to an irregular figure for thepurpose of causing the working length of the beam to correspond thereto;and Figs. 8 and 9 are perspective views of the operators hands and theplanimeter in different positions taken during the operation of theinstrument, the figure to be measured and the material on which itappears being also shown.

In all the figures the same letters refer to the same or correspondingparts.

a is the beam.

1) is the hatchet.

c is the hatchet leg socket.

The spring friction lever arrangement consists of the hinged lever (land the flat spring e, the short arm of the former being kept presseddown against the beam by the upward force of the spring 6 011 the longerarm. On pulling down the longer arm the friction on the beam is relievedand the socket can be readily moved along (2'. c. adjusted on) the beam.

f is the tracer which may be permanently fixed to the beam as shown inthe drawings, or it may be attached thereto by an adjustable socket in asimilar manner to that shown for the hatchet.

In using the planimeter it is of importance to hold the tracer end veryfreely, so that the pressure of the hand always acts through thevertical axis of the tracer, which insures that no twisting motion isimparted to the beam except such as arises from the motion of the tracerover the boundary line and over the line to and from the starting point,from and to the boundary line. To facilitate the use of the instrumentin this manner, a handleg is provided having a conical socket h fittedto one end of the handle by means of the trunnions or screw center studs2'. The conical socket h fits over the conical part of the tracer leg.The legs of the hatchet and tracer are as shown out half away on theirinner sides at right angles to the length of the beam in order tofacilitate the accurate adjustmentof the length of the beam which forthe purpose of this calculation must be understood as approximately thedistance between the central longitudinal lines of the tracer andhatchet which by adjustment of the latter is made to correspond with thelength of the figure to be measured.

As an illustration of the mode of using my improved plauimeter I willdescribe its application to ascertaining the mean or average pressurefrom a steam engine indicator diagram. The instrument is set so that thedis tance between the inside of the legs is equal to the length of theindicator card. A point a, see Fi 4, is chosen near the center of thecard (by eye) and a line a Z) drawn from it to the boundary. The card ispinned down to the table, and the tracing leg of the instrument held inthe right hand, the point being placed at a and the hatchet at w, z'. 6.with the instrument roughly square with a b. The hatchet is pressed inorder to make a slight dent in the paper at :0. The finger having beenremoved from the hatchet, the tracing point of the instrument is causedto traverse the line ab and the boundary b e (Z c b, returning toacid 1) a. It will now now be found that the hatchet has taken up a newposition, and it is again lightly pressed in order to make a fresh dentin the paper (at y). The instrument being held in this position theindicator card is revolved through about one hundred and eighty degrees(by eye) using the point of the instrument as a center, care being takenthat neither the point nor the hatchet shift while the card is beingturned. a Z) will again be roughly at right angles to the beambut in areversed position as indicated by the dotted line A B, as also thedotted lines and capital letters indicate the reversed position of thefigure. The tracing point is now caused to traverse the boundary (indotted lines) as before but in the opposite direction viz:A B E D O B A.The hatchet will take up the new position X which may or may notcoincide with 00. The mean of 001/ and X1 is the main height of thecard. It is well to aim at getting the mean position of the instrument(2'. e. roughly at right angles with a I) rather than making it so tostart with. The hatchet should not be allowed to work on a roughsurface, wood especially is unsuitable as the grain of the wood tends tomake the instrument move along it.

The theory of my instrument maybe demonstrated as follows.

Let 0 B C be the closed curve moved round by the tracing point, and a:and y the (0 ordinates of any position of the tracing point, 5,1; thecoordinates of the corresponding position of the hatchet. It is evidentthat the hatchet can only slide along the line joining my to 5, v Thepoint 5, therefore has no motion or velocity perpendicular to the linejoining my to E, 17; that is a. dv+dy Cos v(Zx Sin 11:0 (1) a being thelength of the instrument.

E-x a Cos v.

fly=aSin v.

Differentiating thelast two equations, we get d:d:v-a. Sin c. (In. and

r dn:rly+a. Cos 11. (In. Therefore 1; (lg-.3 d :(y-{-ct. Sin 1:) (rm-wt.Sin 'U. rlv)--(a'-l rt.

Cos 2:. (In) dy a rZv-l-a ((1.11. Sin 11-1111. Cos n)-a. dv (1 Sin 11+(105 v) (dy-I-a From equation the third and fourth terms of this lastequation equals 0. Therefore fig]. dx-x. dy)= f (n. as 5. dew/ Sin n+2:Cos 0) do.

Now if the tracing point travels completely round the curve(y.dx--x.dy)=2 (area of curve OBC) suppose=2 A.:

but 5 and a do not return to their original position, but the hatchetoccupies a position as Q, suppose P be the original position, so thatPNQ is the glissate traced out. Let the angle QOP:a. Then taken betweenthe limits P and Q -/'(n.dE-E.dn)=2 area OPNQ. :fZ area PQN+2 area QOIwhere QLP is the arc struck with OP (or 0Q) as radius.

Integrating by parts, we obtain.

find (Sin 1;) =:v Sin ii Sin uda: and

1/2 41 (Cos v):-y Cos 0+ f Cos udu.

(a: Cos 'u-i-y Sin 1)) dv=(a: Sin v-y Cos v) f f (Sin mix-Cos 1; dy)

and by substituting from equation :2: Sin v.y Cos v.-a dv and takingthis between the limits corresponding to P and Q Now I have found byexperiment that when the tracing point is made to follow the boundary asabove described, the value of A is two and one-half per cent. of thearea to be measured. Hence I alter the length of the instrument to thisextent, so that still keeping the same scale, no correction need be madefor A.

In the Stang planimeter (by. Cornelius Knudsen of Copenhagen) theposition of the instrument with regard to the line a b was notspecified. Consequently the value of A varied. The value of A had to becalculated by a long and tedious process which rendered the instrumentpractically useless to any but first-rate mathematicians. In the lastequation above therefore, if the length of the instrument (a) is madetwo and one-half per cent. longer than the diagram, then A may beeliminated from this equation and we obtain A=a.arc PQ.

Having now particularly described and ascertained the nature of my saidinvention and in what manner the same is to be employed, I claim as myinvention and desire to secure by Letters Patent- I. In a planimeter,the combination of a rigid beam witha tracer and a hatchet mountedthereon the said tracer being provided with a point for following thelines of the figure to be traced, the said hatchet being provided withan edge which is curved as described for allowing it toturn whileconfining the said hatchet to certain lines of motion and one of theaforesaid parts being adjustable lengthwise of the said beam to suit thelength of the figure substantially as set forth.

2. In a planimeter the combination of a rigid beam with a hatchet andtracer mounted :mean height of diagram=arc PQ.

' thereon the said hatchet being brought to an edge and the said tracerto a point as described and the legs of the said tracer and hatchetbeing cut away approximately to their central line substantially as setforth.

3. In a planimeter the combination of a V and a hatchet mounted on thesaid beam the former being provided with a point and the latter with anedge for bearing on the paper in combination with a handle g and aconical socket h and trunnions 11 for connecting the said lever to thesaid tracer leg substantially as set forth.

JOHN GOODMAN. Witnesses:

HENRY LABHART, ADAM LEHR.

